272 X EXISTENCE THEOREMS

(10.2.2) Letfi be n scalar functions defined and continuously differentiable in
a neighborhood UxVofapoint (a1,...,am,bl,...,bn)ofExF, such that

ftfa* •••> tfm, *i, ••• A) = 0/or 1 Ŧ/< a, and that thejacobian (fl* " '*fn\

v(yi9 • • • > y<n)

is not 0 at (al9..., am ,bly..., bn\ Then there is an open neighborhood W0 c U
°f (ai> • - •ŧO JWC^ ^a*> /or ^^ connected open neighborhood W c W0 0/
(#!,..., flm), /Aere is a unique system ofn scalar functions gt (I < z < Ŧ), defined
and continuous in W #wd .swc/z fAfl/ ^i(al9..., am) = Z>f /or 1 < i ^ w, #/zrf

/,(*!, ..., *w, ^(^, ..., xw),..., gH(xt,..., xj) = 0

/or 1 < i < w aw^ flf/iy (jcl5..., jcw) e W. Moreover, the functions g{ are con-
tinuously differentiable in W, and the jacobian matrix (Dj-g^x)) is equal to
—B~~1A, where A (resp. B) is obtained by replacing yt by gt(xl:>..., xm)
(!</<•Ŧ) in thejacobian matrix (dfjdxk) (resp. (dfjdyjf).

(10.2.3) If the assumptions of (10.2.1) are verified, and if in addition f is p
times continuously differentiable in a neighborhood of (x0, yo), then u is p
times continuously differentiable in a neighborhood of XQ .

We prove by induction on k that w is A: times continuously differentiable
for 1 < k </?; for k = 1, this follows from (10.2.1), and moreover u'(x) =
F(x,u(x))9 where F(x,y) = - (Da/foj;))-1 ° (Dąf(x9y)) is p - 1 times
continuously differentiable by (8.12.9), (8.12.11), and (8.12.10). By (8.12.10),
u' is therefore k — 1 times continuously differentiable (for k ^ p) and that
means that u is k times continuously differentiable by (8.12.5).

(10.2.4) Suppose E, F, G are finite dimensional, and f is analytic in A; then
u is analytic in a neighborhood of XQ .

If the field of scalars K is C, the result follows from (10.2.1) and the
characterization of analytic functions as continuously differentiable functions
(9.10.1). Suppose now K = R, E = Rm, F = G = Rn; then there is an open set
B c Cm+n such that B n Rm+/1 = A and an analytic mapping g of B into
Cn which extends/(9.4.5). Identifying D2/and D2g with jacobian matrices
shows that ~D2g(xQ, y0) transforms a basis of Rn over R into a basis of Rn, and
these bases are also bases of C" over C, hence D2#(jc0, yQ) is a linear homeo-
morphism of C1 onto itself. We therefore can apply (10.2.1) to g, which shows
the existence of an analytic mapping z; of a neighborhood W of x0 in Cm such
that g(z, v(z)) = 0 and v(x0) = y0. Moreover, it follows from formula (10.2.1.1)
by induction on |v| that all the derivatives Dvi? at the point JCQ map Rm into2\foTxel9zl9z2 in H (compare to (10.1.1)); (2) for